1Complex numbers

IA Vectors and Matrices

1.4 Complex logarithm and power

Definition

(Complex logarithm)

.

The complex logarithm

w

=

log z

is a solution

to

e

ω

=

z

, i.e.

ω

=

log z

. Writing

z

=

re

iθ

, we have

log z

=

log

(

re

iθ

) =

log r

+

iθ

.

This can be multi-valued for different values of

θ

and, as above, we should select

the θ that satisfies −π < θ ≤ π.

Example. log 2i = log 2 + i

π

2

Definition

(Complex power)

.

The complex power

z

α

for

z, α ∈ C

is defined as

z

α

=

e

α log z

. This, again, can be multi-valued, as

z

α

=

e

α log |z|

e

iαθ

e

2inπα

(there

are finitely many values if

α ∈ Q

, infinitely many otherwise). Nevertheless, we

make z

α

single-valued by insisting −π < θ ≤ π.